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G = C42.665C23order 128 = 27

80th non-split extension by C42 of C23 acting via C23/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.665C23, (C2×C8).34D4, C8⋊C8.5C2, C4.8(C4○D8), C83Q8.8C2, C82Q8.9C2, C4⋊Q8.89C22, (C4×C8).258C22, C2.8(C8.2D4), C4.3(C8.C22), C4.SD16.6C2, C2.13(C8.12D4), C22.66(C41D4), (C2×C4).722(C2×D4), SmallGroup(128,450)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C42.665C23
C1C2C22C2×C4C42C4×C8C8⋊C8 — C42.665C23
C1C22C42 — C42.665C23
C1C22C42 — C42.665C23
C1C22C22C42 — C42.665C23

Generators and relations for C42.665C23
 G = < a,b,c,d,e | a4=b4=1, c2=a2, d2=ab2, e2=b, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=a-1c, ece-1=b-1c, ede-1=a2d >

Subgroups: 176 in 82 conjugacy classes, 36 normal (24 characteristic)
C1, C2 [×3], C4 [×6], C4 [×4], C22, C8 [×6], C2×C4 [×3], C2×C4 [×4], Q8 [×8], C42, C4⋊C4 [×8], C2×C8 [×6], C2×Q8 [×4], C4×C8 [×3], Q8⋊C4 [×8], C4.Q8 [×2], C2.D8 [×2], C4⋊Q8 [×4], C8⋊C8, C4.SD16 [×4], C83Q8, C82Q8, C42.665C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C41D4, C4○D8 [×2], C8.C22 [×4], C8.12D4, C8.2D4 [×2], C42.665C23

Character table of C42.665C23

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J8K8L
 size 111122222216161616444444444444
ρ111111111111111111111111111    trivial
ρ211111111111-11-1-1-1-1-11-1-11-1-111    linear of order 2
ρ31111111111-1-11111-1-1-11-1-11-1-1-1    linear of order 2
ρ41111111111-111-1-1-111-1-11-1-11-1-1    linear of order 2
ρ51111111111-1-1-1-1111111111111    linear of order 2
ρ61111111111-11-11-1-1-1-11-1-11-1-111    linear of order 2
ρ7111111111111-1-111-1-1-11-1-11-1-1-1    linear of order 2
ρ811111111111-1-11-1-111-1-11-1-11-1-1    linear of order 2
ρ92222-22-2-22-200000000200-200-22    orthogonal lifted from D4
ρ1022222-22-2-2-200002-2000-2002000    orthogonal lifted from D4
ρ112222-2-2-22-22000000-2-200200200    orthogonal lifted from D4
ρ122222-2-2-22-220000002200-200-200    orthogonal lifted from D4
ρ132222-22-2-22-200000000-2002002-2    orthogonal lifted from D4
ρ1422222-22-2-2-20000-22000200-2000    orthogonal lifted from D4
ρ152-22-20-20020000022-2--20-2--22i-2-2-2i0    complex lifted from C4○D8
ρ162-22-20-200200000-2-2--2-202-22i2--2-2i0    complex lifted from C4○D8
ρ172-22-20200-200000-22-2--22i-2-202--20-2i    complex lifted from C4○D8
ρ182-22-20-200200000-2-2-2--202--2-2i2-22i0    complex lifted from C4○D8
ρ192-22-20200-2000002-2-2--2-2i2-20-2--202i    complex lifted from C4○D8
ρ202-22-20-20020000022--2-20-2-2-2i-2--22i0    complex lifted from C4○D8
ρ212-22-20200-2000002-2--2-22i2--20-2-20-2i    complex lifted from C4○D8
ρ222-22-20200-200000-22--2-2-2i-2--202-202i    complex lifted from C4○D8
ρ234-4-4440-40000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2444-4-4000-4040000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-4-44-4040000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2644-4-400040-40000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C42.665C23
Regular action on 128 points
Generators in S128
(1 56 5 52)(2 49 6 53)(3 50 7 54)(4 51 8 55)(9 121 13 125)(10 122 14 126)(11 123 15 127)(12 124 16 128)(17 27 21 31)(18 28 22 32)(19 29 23 25)(20 30 24 26)(33 42 37 46)(34 43 38 47)(35 44 39 48)(36 45 40 41)(57 98 61 102)(58 99 62 103)(59 100 63 104)(60 101 64 97)(65 80 69 76)(66 73 70 77)(67 74 71 78)(68 75 72 79)(81 107 85 111)(82 108 86 112)(83 109 87 105)(84 110 88 106)(89 120 93 116)(90 113 94 117)(91 114 95 118)(92 115 96 119)
(1 61 54 100)(2 62 55 101)(3 63 56 102)(4 64 49 103)(5 57 50 104)(6 58 51 97)(7 59 52 98)(8 60 53 99)(9 105 127 81)(10 106 128 82)(11 107 121 83)(12 108 122 84)(13 109 123 85)(14 110 124 86)(15 111 125 87)(16 112 126 88)(17 67 25 80)(18 68 26 73)(19 69 27 74)(20 70 28 75)(21 71 29 76)(22 72 30 77)(23 65 31 78)(24 66 32 79)(33 93 48 114)(34 94 41 115)(35 95 42 116)(36 96 43 117)(37 89 44 118)(38 90 45 119)(39 91 46 120)(40 92 47 113)
(1 109 5 105)(2 88 6 84)(3 107 7 111)(4 86 8 82)(9 61 13 57)(10 103 14 99)(11 59 15 63)(12 101 16 97)(17 94 21 90)(18 118 22 114)(19 92 23 96)(20 116 24 120)(25 115 29 119)(26 89 30 93)(27 113 31 117)(28 95 32 91)(33 73 37 77)(34 71 38 67)(35 79 39 75)(36 69 40 65)(41 76 45 80)(42 66 46 70)(43 74 47 78)(44 72 48 68)(49 110 53 106)(50 81 54 85)(51 108 55 112)(52 87 56 83)(58 122 62 126)(60 128 64 124)(98 125 102 121)(100 123 104 127)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)
(1 76 61 21 54 71 100 29)(2 73 62 18 55 68 101 26)(3 78 63 23 56 65 102 31)(4 75 64 20 49 70 103 28)(5 80 57 17 50 67 104 25)(6 77 58 22 51 72 97 30)(7 74 59 19 52 69 98 27)(8 79 60 24 53 66 99 32)(9 34 105 94 127 41 81 115)(10 39 106 91 128 46 82 120)(11 36 107 96 121 43 83 117)(12 33 108 93 122 48 84 114)(13 38 109 90 123 45 85 119)(14 35 110 95 124 42 86 116)(15 40 111 92 125 47 87 113)(16 37 112 89 126 44 88 118)

G:=sub<Sym(128)| (1,56,5,52)(2,49,6,53)(3,50,7,54)(4,51,8,55)(9,121,13,125)(10,122,14,126)(11,123,15,127)(12,124,16,128)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26)(33,42,37,46)(34,43,38,47)(35,44,39,48)(36,45,40,41)(57,98,61,102)(58,99,62,103)(59,100,63,104)(60,101,64,97)(65,80,69,76)(66,73,70,77)(67,74,71,78)(68,75,72,79)(81,107,85,111)(82,108,86,112)(83,109,87,105)(84,110,88,106)(89,120,93,116)(90,113,94,117)(91,114,95,118)(92,115,96,119), (1,61,54,100)(2,62,55,101)(3,63,56,102)(4,64,49,103)(5,57,50,104)(6,58,51,97)(7,59,52,98)(8,60,53,99)(9,105,127,81)(10,106,128,82)(11,107,121,83)(12,108,122,84)(13,109,123,85)(14,110,124,86)(15,111,125,87)(16,112,126,88)(17,67,25,80)(18,68,26,73)(19,69,27,74)(20,70,28,75)(21,71,29,76)(22,72,30,77)(23,65,31,78)(24,66,32,79)(33,93,48,114)(34,94,41,115)(35,95,42,116)(36,96,43,117)(37,89,44,118)(38,90,45,119)(39,91,46,120)(40,92,47,113), (1,109,5,105)(2,88,6,84)(3,107,7,111)(4,86,8,82)(9,61,13,57)(10,103,14,99)(11,59,15,63)(12,101,16,97)(17,94,21,90)(18,118,22,114)(19,92,23,96)(20,116,24,120)(25,115,29,119)(26,89,30,93)(27,113,31,117)(28,95,32,91)(33,73,37,77)(34,71,38,67)(35,79,39,75)(36,69,40,65)(41,76,45,80)(42,66,46,70)(43,74,47,78)(44,72,48,68)(49,110,53,106)(50,81,54,85)(51,108,55,112)(52,87,56,83)(58,122,62,126)(60,128,64,124)(98,125,102,121)(100,123,104,127), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,76,61,21,54,71,100,29)(2,73,62,18,55,68,101,26)(3,78,63,23,56,65,102,31)(4,75,64,20,49,70,103,28)(5,80,57,17,50,67,104,25)(6,77,58,22,51,72,97,30)(7,74,59,19,52,69,98,27)(8,79,60,24,53,66,99,32)(9,34,105,94,127,41,81,115)(10,39,106,91,128,46,82,120)(11,36,107,96,121,43,83,117)(12,33,108,93,122,48,84,114)(13,38,109,90,123,45,85,119)(14,35,110,95,124,42,86,116)(15,40,111,92,125,47,87,113)(16,37,112,89,126,44,88,118)>;

G:=Group( (1,56,5,52)(2,49,6,53)(3,50,7,54)(4,51,8,55)(9,121,13,125)(10,122,14,126)(11,123,15,127)(12,124,16,128)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26)(33,42,37,46)(34,43,38,47)(35,44,39,48)(36,45,40,41)(57,98,61,102)(58,99,62,103)(59,100,63,104)(60,101,64,97)(65,80,69,76)(66,73,70,77)(67,74,71,78)(68,75,72,79)(81,107,85,111)(82,108,86,112)(83,109,87,105)(84,110,88,106)(89,120,93,116)(90,113,94,117)(91,114,95,118)(92,115,96,119), (1,61,54,100)(2,62,55,101)(3,63,56,102)(4,64,49,103)(5,57,50,104)(6,58,51,97)(7,59,52,98)(8,60,53,99)(9,105,127,81)(10,106,128,82)(11,107,121,83)(12,108,122,84)(13,109,123,85)(14,110,124,86)(15,111,125,87)(16,112,126,88)(17,67,25,80)(18,68,26,73)(19,69,27,74)(20,70,28,75)(21,71,29,76)(22,72,30,77)(23,65,31,78)(24,66,32,79)(33,93,48,114)(34,94,41,115)(35,95,42,116)(36,96,43,117)(37,89,44,118)(38,90,45,119)(39,91,46,120)(40,92,47,113), (1,109,5,105)(2,88,6,84)(3,107,7,111)(4,86,8,82)(9,61,13,57)(10,103,14,99)(11,59,15,63)(12,101,16,97)(17,94,21,90)(18,118,22,114)(19,92,23,96)(20,116,24,120)(25,115,29,119)(26,89,30,93)(27,113,31,117)(28,95,32,91)(33,73,37,77)(34,71,38,67)(35,79,39,75)(36,69,40,65)(41,76,45,80)(42,66,46,70)(43,74,47,78)(44,72,48,68)(49,110,53,106)(50,81,54,85)(51,108,55,112)(52,87,56,83)(58,122,62,126)(60,128,64,124)(98,125,102,121)(100,123,104,127), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,76,61,21,54,71,100,29)(2,73,62,18,55,68,101,26)(3,78,63,23,56,65,102,31)(4,75,64,20,49,70,103,28)(5,80,57,17,50,67,104,25)(6,77,58,22,51,72,97,30)(7,74,59,19,52,69,98,27)(8,79,60,24,53,66,99,32)(9,34,105,94,127,41,81,115)(10,39,106,91,128,46,82,120)(11,36,107,96,121,43,83,117)(12,33,108,93,122,48,84,114)(13,38,109,90,123,45,85,119)(14,35,110,95,124,42,86,116)(15,40,111,92,125,47,87,113)(16,37,112,89,126,44,88,118) );

G=PermutationGroup([(1,56,5,52),(2,49,6,53),(3,50,7,54),(4,51,8,55),(9,121,13,125),(10,122,14,126),(11,123,15,127),(12,124,16,128),(17,27,21,31),(18,28,22,32),(19,29,23,25),(20,30,24,26),(33,42,37,46),(34,43,38,47),(35,44,39,48),(36,45,40,41),(57,98,61,102),(58,99,62,103),(59,100,63,104),(60,101,64,97),(65,80,69,76),(66,73,70,77),(67,74,71,78),(68,75,72,79),(81,107,85,111),(82,108,86,112),(83,109,87,105),(84,110,88,106),(89,120,93,116),(90,113,94,117),(91,114,95,118),(92,115,96,119)], [(1,61,54,100),(2,62,55,101),(3,63,56,102),(4,64,49,103),(5,57,50,104),(6,58,51,97),(7,59,52,98),(8,60,53,99),(9,105,127,81),(10,106,128,82),(11,107,121,83),(12,108,122,84),(13,109,123,85),(14,110,124,86),(15,111,125,87),(16,112,126,88),(17,67,25,80),(18,68,26,73),(19,69,27,74),(20,70,28,75),(21,71,29,76),(22,72,30,77),(23,65,31,78),(24,66,32,79),(33,93,48,114),(34,94,41,115),(35,95,42,116),(36,96,43,117),(37,89,44,118),(38,90,45,119),(39,91,46,120),(40,92,47,113)], [(1,109,5,105),(2,88,6,84),(3,107,7,111),(4,86,8,82),(9,61,13,57),(10,103,14,99),(11,59,15,63),(12,101,16,97),(17,94,21,90),(18,118,22,114),(19,92,23,96),(20,116,24,120),(25,115,29,119),(26,89,30,93),(27,113,31,117),(28,95,32,91),(33,73,37,77),(34,71,38,67),(35,79,39,75),(36,69,40,65),(41,76,45,80),(42,66,46,70),(43,74,47,78),(44,72,48,68),(49,110,53,106),(50,81,54,85),(51,108,55,112),(52,87,56,83),(58,122,62,126),(60,128,64,124),(98,125,102,121),(100,123,104,127)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128)], [(1,76,61,21,54,71,100,29),(2,73,62,18,55,68,101,26),(3,78,63,23,56,65,102,31),(4,75,64,20,49,70,103,28),(5,80,57,17,50,67,104,25),(6,77,58,22,51,72,97,30),(7,74,59,19,52,69,98,27),(8,79,60,24,53,66,99,32),(9,34,105,94,127,41,81,115),(10,39,106,91,128,46,82,120),(11,36,107,96,121,43,83,117),(12,33,108,93,122,48,84,114),(13,38,109,90,123,45,85,119),(14,35,110,95,124,42,86,116),(15,40,111,92,125,47,87,113),(16,37,112,89,126,44,88,118)])

Matrix representation of C42.665C23 in GL6(𝔽17)

1600000
0160000
0011600
0021600
0012440
0072013
,
010000
1600000
0016000
0001600
0000160
0000016
,
640000
4110000
0013131410
0011224
001441211
00811014
,
0130000
400000
000001
00101505
00152213
0011600
,
5120000
550000
000010
00513140
0016000
00215154

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,2,12,7,0,0,16,16,4,2,0,0,0,0,4,0,0,0,0,0,0,13],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[6,4,0,0,0,0,4,11,0,0,0,0,0,0,13,1,14,8,0,0,13,12,4,1,0,0,14,2,12,10,0,0,10,4,11,14],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,10,15,1,0,0,0,15,2,16,0,0,0,0,2,0,0,0,1,5,13,0],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,5,16,2,0,0,0,13,0,15,0,0,1,14,0,15,0,0,0,0,0,4] >;

C42.665C23 in GAP, Magma, Sage, TeX

C_4^2._{665}C_2^3
% in TeX

G:=Group("C4^2.665C2^3");
// GroupNames label

G:=SmallGroup(128,450);
// by ID

G=gap.SmallGroup(128,450);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,512,422,387,100,1123,136,2804,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2,d^2=a*b^2,e^2=b,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=a^-1*c,e*c*e^-1=b^-1*c,e*d*e^-1=a^2*d>;
// generators/relations

Export

Character table of C42.665C23 in TeX

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